Means of Hilbert Space Operators

Hiai, Fumio; Kosaki, Hideki

doi:10.1007/b13213

Cited by 71 publications

(75 citation statements)

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“…We presented the proof here because it is short and "coordinate free". For functions that define DOI on S ∞ (or, equivalently, on S 1 , S b ), a precise description was obtained by Peller [1985], completing previous results of Birman and Solomyak [1967;1973] (a transparent proof of Peller's theorem can be found in the recent book [Hiai and Kosaki 2003]). Without stating this directly, Peller's theorem shows that only supports of the spectral measures are essential in the description of M π,ρ .…”

Section: Multipliers Of Commutative Algebras; (µ ν)-Multiplierssupporting

confidence: 63%

Operator multipliers

Kissin

Shulman

2006

Pacific J. Math.

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We introduce a noncommutative version of Schur multipliers relative to an operator ideal. In this setting the functions of two variables are replaced by elements from a tensor product of C*-algebras, and the measures (or spectral measures) by representations. For commutative C*-algebras this approach agrees with Birman and Solomyak's theory of double operator integrals. We study the dependence of the spaces of multipliers on the choice of representations and find that the question is closely related to Voiculescu and Arveson's theory of approximately equivalent representations. The space of multipliers universal with respect to the chosen measures is related to the Haagerup tensor product of the algebras.

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Section: Multipliers Of Commutative Algebras; (µ ν)-Multiplierssupporting

confidence: 63%

Operator multipliers

Kissin

Shulman

2006

Pacific J. Math.

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Section: Multipliers Of Commutative Algebras; (µ ν)-Multiplierssupporting

confidence: 60%

Untitled

2006

Pacific J. Math.

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“…For a function f the corresponding mean is defined as m f (x, y) = f (x/y)y for positive numbers, or for commuting positive definite matrices. (In fact, if f : R + → R + satisfies f (1) = 1 and certain other conditions, then m f is a mean discussed in [9].) The linear mappings…”

Section: Quasi-entropymentioning

confidence: 99%

From Quasi-Entropy to Various Quantum Information Quantities

Hiai¹,

Petz²

2012

Publ. Res. Inst. Math. Sci.

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The subject is the applications of the use of quasi-entropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the f -divergence, monotone metrics, covariance and the χ 2 -divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.2000 Mathematics Subject Classification. Primary 81P45; Secondary 54C70.

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Means of Hilbert Space Operators

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 71 publications

References 0 publications

Operator multipliers

Operator multipliers

Untitled

From Quasi-Entropy to Various Quantum Information Quantities

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